Although @Bastien Montagne (mont29) mentioned in T51543 that this sort of behaviour is apparently how it's now supposed to work, I'll take a look at what's going on here, as the current behaviour doesn't look very useful at all.

I think Flip names does what it should do currently - flipping names, not trying to change names to match (and usually fail in all but simplest cases) some mirror pattern.

IMHO, here we'd rather need a new 'Copy Mirror' operator, that would properly duplicate and rename bones at the same time? I cannot really see how we could achieve a proper 'name mirroring' a posteriori anyways, too much corner cases that would break such attempt.

Symmetrize doesn't necessarily solve the problem, as it needs a root bone/object to symmetrize against. See my .blend on T52671 for a case where Symmetrize wouldn't work at all.

I feel like T50393 was the corner case (selecting both sides of an armature and doing flip names? Who does that? And WHY? And Dalai had dismissed it as not a bug at the time) and fixing it broke the standard use case. Flip names worked the way it worked at least since 2.49, and nobody minded.